A bicycle wheel is mounted on a fixed, frictionless axle, with alight string wound around its rim. The wheel has moment…
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Question “A bicycle wheel is mounted on a fixed, frictionless axle, with alight string wound around its rim. The wheel has moment…”
A bicycle wheel is mounted on a fixed, frictionless axle, with alight string wound around its rim. The wheel has moment of inertia I=kmr2, where m is its mass, r is its radius, and k is a dimensionless constant between zero and one. The wheel is rotating counterclockwise with angular velocity w0, when at time t=0 someone starts pulling the string with a force of magnitude F. Assume that the string does not slip on the wheel.
Part A
Suppose that after a certain time tL, the string has been pulled through a distance L. What is the final rotational speed wfinal of the wheel?
Express your answer in terms of L, F, I, and w0.
Part B
What is the instantaneous power P delivered to the wheel via the force F at time t=0?
Express the power in terms of some or all of the variables given in the problem introduction.
Answer
Concepts and Reason
These concepts are required to answer these questions: rotational kinetic energie, work energy theorem and torque.
First, calculate the final and initial kinetic energy. Then use the work-energy theorem for the change in the rotational kinetic energies. Find the final angular velocity by solving this equation. Next, calculate torque on the wheel due to force. Then calculate the power delivered.
Fundamentals
The energy derived from an object’s rotational motion is called the rotational kinetic energy. If the object’s moment of inertia and its angular velocity is then its rotational energy is.
work energytheorem says that the work done to an object is in the form change in its kinetic energie. represents the object’s initial kinetic energie, the object’s final kineticenergy, and the amount of work performed is.
The force that causes a change in the rotational state is called torque. It is the product of force applied and position vector from the origin at the center of rotation to the point at which it acts.
The magnitude of torque can be expressed as:
refers to the angle between force vector and position vector.
Work can be defined as the product of force and its corresponding displacement in force’s direction.
indicates the distance.
By angle
is.
The expression for power in rotational motion is
P refers to the power,
the torque and
the angular velocity.
(A)
For initial rotational kinetic energie of the wheel, it is:
represents the initial angular velocity.
The final kinetic energy is the following:
The expression for the work energy theorem can be found at,
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the object’s final kineticenergy, and the amount of work performed is.
